μ-Complete Souslin Trees on μ^{+}

by Menachem Kojman and Saharon Shelah† The Hebrew University, Jerusalem ABSTRACT We prove that μ = μ^{<μ}, 2^{μ} = μ^{+} and “there is a non reflecting stationary subset of μ^{+}

composed of ordinals of cofinality < μ” imply that there is a μ-complete Souslin tree on μ^{+}.

Introduction The old problem of the existence of Souslin trees has attracted the attention of many (see [Je] for history). While the ℵ_{1} case is settled, the consistency of GCH + SH(ℵ_{2}) is still an open question. Gregory showed in [G] that GCH + “there is a non reflecting stationary set of ω-cofinal elements of ω_{2}” implies the existence of an ℵ_{2}-Souslin tree. Gregory’s result showed that the consistency strength of GCH + SH(ℵ_{2}) is at least that of the existence of a Mahlo cardinal. Without GCH, the consistency of CH + SH(ℵ_{2}) is known from [LvSh 104]. In [ShSt 279] the equiconsistency of the existence of a weakly compact cardinal with “every ℵ_{2}-Aronszajn tree is special” is shown. In [ShSt 154] it is shown that under CH, the consistency strength of “there are no ℵ_{1}-complete ℵ_{2}-Souslin trees” is at least that of an inaccessible cardinal.

We show how a Souslin tree which is μ-complete (μ regular) can be constructed on a cardinal μ^{+} from a certain combinatorial principle (Theorem 2 below), and then show how this principle may be gotten from GCH and a non reflecting stationary set of ordinals with cofinality < μ in μ^{+} (Theorem 3 below). As a corollary (Corollary 5 below), GCH + “there is a non reflecting stationary set of ω-cofinal elements of ω_{2}” implies the existence of an ℵ_{1}-complete Souslin tree on ℵ_{2}.

As mentioned in [G], 1.10(3), CH and the existence of a diamond sequence on {δ < ℵ_{2} : cfδ = ℵ_{1}} imply the existence of a Souslin tree on ℵ_{2} which is ℵ_{1}-complete. The construction of such a tree is by induction on levels, where at a stage of countable cofinality all branches are realized (for ℵ_{1}-completeness), while at stages of cofinality ℵ_{1} the diamond is consulted to realize only a part of the cofinal branches in a way which kills all future big antichains. The combinatorial principle used in Theorem 2 to construct a μ-complete Souslin tree on μ^{+} under GCH can be viewed as a weaker substitute for a diamond sequence on {δ < μ^{+} : cfδ = μ}: instead of using a guess at a single guessat stage of cofinality μ, we use unboundedly many guesses, each at a level of cofinality < μ.

Unlike a diamond sequence on the stationary set of critical cofinality, this principle makes sense also in the case of an inaccessible cardinal (where there is no “critical cofinality”). This principle is closely related to club guessing (see [Sh-g] and [Sh-e]), which was discovered while the second

† The second author thanks the Binational Science Foundation for supporting this research. Publication no. 449

1 Souslin 27.9.1998 author was trying to prove some results in Model Theory. This principle continues the principle that appears in [AbShSo 221], in which Souslin trees on successors of singulars are treated.

We learned from the referee that Gregory presented in the seventies in a seminar at Buffalo a construction of a countably complete Souslin tree on ℵ_{2} from GCH and a square, but that this was not written.

1. Notation: : (1) If C is a set of ordinals, then acc C is the set of accumulation points of C and nacc C =^{df} C \ acc C. By T_{α} we denote the α-th level of the tree T and by T (α) we denote ⋃_{b<α} T_{β}.

2. Theorem: Suppose that

(a) λ = μ^{+} = 2^{μ}, μ = μ^{<μ};

(b) S^{∗} ⊆ \{α ∈ λ : cfα = μ\} and \overline{C} = ⟨c_{δ} : δ ∈ S^{∗}\}, δ = sup c_{δ}, c_{δ} is a closed set of limit ordinals.

(c) For every δ ∈ S^{∗} and α ∈ nacc c_{δ}, P_{δ,α} ⊆ P(α), |P_{δ,α}| ≤ cfα, and if α ∈ S^{∗}, then |P_{δ,α}| < μ;

(d) For every set A ⊆ λ and club E ⊆ λ, there is a stationary S_{A,E} ⊆ S^{∗} such that for every δ ∈ S_{A,E}, δ = sup{α ∈ nacc c_{δ} : A ∩ α ∈ P_{δ,α} ∧ α ∈ E};

(e) If δ, δ^{∗} ∈ S^{∗}, δ ∈ acc c_{δ^{∗}}, then there is some α < δ such that ⟨P_{δ,β} : β ∈ nacc c_{δ} ∧ β > α⟩ = ⟨P_{δ^{∗},β} : β ∈ (nacc c_{δ} ∩ δ) ∧ β > α⟩.

(f) for every γ < λ, |{⟨P_{δ,α} : α ∈ C_{δ} ∩ γ⟩ : δ ∈ S}| ≤ μ. Then there is a μ-complete Souslin tree on λ.

Discussion: Condition (d) is the prediction demand. It says that for every club E and a set A there is a stationary set of δ-s, such that for unboundedly many non-accumulation points α of c_{δ} two things happen: α ∈ E it and A ∩ α is guessed by P_{δ,α}. Proof: We assume, without loss of generality, that for every δ ∈ S^{∗} and α ∈ c_{δ}, α = μα. By induction on α < λ we construct a tree T (α) of height α such that:

(i) The universe of T (α) is μ(α+1), the β-th level in T (α), T_{β}, consists of the elements [μβ, μ(β + 1)), and every x ∈ T_{β} for β < α has an extension in T_{γ} for every γ < α. Every x ∈ T (α) such that Lev(x) + 1 < α has at least two immediate successors in T_{α}.

(ii) T (α) is μ-complete; (iii) For α < β, T (α) = T (β) |T (α)|

Also, we define a partial function (which, intuitively speaking, chooses branches which help us in preserving the maximality of small antichains that occur along the way): (iv) For every x ∈ T (α) and a sequence t = ⟨P_{δ,β} : β ∈ nacc c_{δ} ∩ α⟩ such that sup(c_{δ} ∩ α) < α and

Lev(x) < max(c_{δ} ∩ α) and max(c_{δ} ∩ α) ∈ nacc c_{δ}, y(x, t) is defined, and is an element in the level max(c_{δ} ∩ α) which extends x and has the property that for every A ∈ P_{δ,max(c_{δ}∩α)} which is a maximal antichain of T (max c_{δ} ∩ α), there is an element of A below y(x, t).

(v) If the sequence s extends the sequence t and y(x, t), y(x, s) exist, then T (α) ⊧ y(x, t) < y(x, s).

(vi) For every increasing sequence ⟨t_{i} : i < i^{∗}⟩ there is an upper bound (in the tree order) to ⟨y(x, t_{i}) : i < i^{∗}⟩. The last demand is:

(vii) If α = δ+1, δ ∈ S^{∗} then every y ∈ T_{δ} satisfies that there is some δ^{∗} ≥ δ ∈ acc c_{δ^{∗}} and x ∈ T (δ), such that y is the least upper bound (in the tree order) of ⟨y(x, t_{α}) : α ∈ nacc c_{δ^{∗}} ∩δ ∧ α_{x} < α < δ⟩ where α_{x} is the least in nacc c_{δ^{∗}} such that α_{x} > Lev(x), and t_{α} = ⟨P_{δ,β} : β ∈ nacc c_{d^{∗}} ∧ β ≤ α⟩. We first show that this construction, once carried out, yields a μ-complete Souslin tree on

λ. The completeness of T = ∪T (α) is clear from the regularity of λ. Suppose that A ⊆ λ is a maximal antichain of T of size λ. Let E be the club of points δ < λ such that T δ = T (δ) and A δ is a maximal antichain of T (δ). Pick a point δ ∈ S^{∗} such that δ = sup{α ∈ nacc c_{δ} : α ∈ E ∧ A α ∈ P_{δ,α}}. As |T (δ)| < λ there is an element a ∈ A, Lev(a) > δ. Let y be the unique such that Lev(y) = δ and y < a. Then by demand (vii), there is some δ^{∗} ≥ δ and x ∈ T (δ) such that y is the least upper bound (in the tree order) of ⟨y(x, t_{α}) : α ∈ nacc c_{δ^{∗}} ∩ δ ∧ α > Lev(x)⟩. There is some α^{∗} < δ such that ⟨P_{δ,β} : α < β < δ ∧ β ∈ nacc c_{δ}⟩ = ⟨P_{δ^{∗},β} : α^{∗} < β ∈ nacc c_{δ^{∗}} ∩ δ⟩. Pick some α ∈ nacc c_{δ} such that α > max{Lev(x), α^{∗}}, α ∈ E and A α ∈ P_{δ,α_{i}}. So α ∈ nacc c_{δ^{∗}} and A ∩ α ∈ P_{δ^{∗},α}. Then the unique x′ < y with Lev(x′) = α (which equals y(x, ⟨P_{δ^{∗},γ} : γ ∈ (nacc c_{δ^{∗}} ∩ (α + 1))⟩)) is above an element a′ ∈ A α. But x′ < a — a contradiction to the fact that A is an antichain.

Next let us show that we can carry out the construction by induction. When α = β + 1 and β is a successor or zero, add two immediate successors to every point in the β-th level. When β is limit, cfβ < μ, add an element above every infinite branch. This addition amounts to the total of μ^{<μ} = μ points. If, in addition, β ∈ nacc c_{δ} for some δ ∈ S^{∗}, then for every x ∈ T (β) define y(x, ⟨P_{δ,γ} : γ ∈ nacc c_{δ} ∧ γ ≤ β⟩) as follows: let γ_{0} = max(c_{δ} ∩ β). When Lev(x) < γ_{0} set x_{0} as the supremum (in T (α)) of ⟨y(x, ⟨P_{δ,α} : α ≤ α^{∗})⟩) : α^{∗} ≤ γ_{0} ∧ α^{∗} ∈ nacc c_{δ}⟩; else, x_{0} = x. As |P_{δ,β}| ≤ cfβ, we can in cfβ steps choose a cofinal branch above x_{0} which has a point above an element from A for every A ∈ P_{δ,β} which is a maximal antichain of T_{β}. Let the required y be the supremum of this branch.

If β is a limit and cfβ = μ, distinguish two cases: case (a): β = δ ∈ S^{∗}. So we should satisfy demand (vii), namely add bounds precisely to those branches which for some δ^{∗} ≥ δ in S^{∗}, δ ∈ acc c_{δ^{∗}}, are of the form ⟨y(x, t_{γ}) : γ ∈ nacc c_{δ^{∗}} ∩β⟩ where t_{γ} = ⟨P_{δ^{∗},ζ} : ζ ≤ γ ∧ζ ∈ nacc c_{δ^{∗}}⟩. By (f) this costs only the addition of μ new elements. If, in addition, there is some δ′ ∈ S^{∗} such that δ ∈ nacc c_{δ′}, we should define y(x, ⟨P_{δ′,γ} : γ ∈ nacc c_{δ′} ∧ γ ≤ δ⟩) for all x ∈ T (δ). This presents no problem: as |P_{δ′,δ}| < μ, we attach to each x some x_{0} such that x_{0} = x or T_{δ} ⊧ x_{0} > x and such that x_{0} is above members from every maximal antichain in P_{δ′,δ}; now y(x, ⟨P_{δ′,γ} : γ ∈ nacc c_{δ′} ∧ γ ≤ δ⟩) will be the point in level δ above x_{0} we obtained anyway to satisfy demand (vii).

Case (b): cfβ = μ and β ∉ S^{∗}. Then when there is some δ such that β ∈ acc c_{δ} we realize enough limits to obtain completeness under increasing sequences of the form ⟨y(x, t_{i}) : i < i^{∗}⟩. By (f), we add thus ≤ μ elements. If there is no such δ, just make sure, by adding μ points to the tree in level β, that above every x ∈ T (β) there is a point in level β. This takes care also of (i). If there is some δ′ such that β ∈ nacc c_{δ′}, then for every x ∈ T (β) define y(x, ⟨P_{δ′,γ} : γ ∈ nacc c_{δ′} ∧ γ ≤ δ⟩) exactly as in the case of smaller cofinality.⊣2

We will show now how to obtain from a non-reflecting stationary set a special case of the prediction principle we used in the previous theorem. One should substitute P_{δ,α} in the previous theorem by B_{α} from the next theorem to get the assumptions of the previous theorem.

3. Theorem: Suppose λ = cfλ > ℵ_{1} , S ⊆ λ is stationary, non-reflecting, and carries a diamond sequence ⟨A_{α} : α ∈ S⟩, S^{∗} is a given non reflecting stationary subset of λ, S^{∗} ∩ S = ∅ and δ ∈ S^{∗} ⇒ cfδ > ℵ_{0}. Then there are sequences \overline{C} = ⟨c_{δ} : δ ∈ S^{∗}⟩ and \overline{B} = ⟨B_{α} : α ∈ S⟩ such that:

(i) B_{α} ⊆ α; (ii) sup c_{δ} = δ and c_{δ} is a closed set of limit ordinals; (iii) if δ, δ^{∗} ∈ S^{∗} and δ ∈ acc c_{δ^{∗}}, then there is some α < δ such that c_{δ^{∗}} ∩ (α, δ) = c_{δ} ∩ (α, δ); (iv) for every club E ⊆ λ and set X ⊆ λ there are stationarily many δ ∈ S^{∗} such that δ = sup{α ∈

nacc c_{δ} : α ∈ S ∩ E ∧ X ∩ α = A_{α}}. Proof: We fix some 1-1 pairing function ⟨−, −⟩ from λ×ω_{0} onto λ and let A_{n}^{α} = {β < α : ⟨β, n⟩ ∈ A\}. We may assume that for every countable sequence \overline{X} = ⟨X_{n} : n < ω⟩ of subsets of λ there are stationarily many α ∈ S such that for every n, X_{n} ∩ α = A_{n}^{α}. Denote by S(\overline{X}), for a (finite or infinite) sequence of subsets of λ the stationary set {α ∈ S : ⋀_{n} X_{n} ∩ α = A_{n}^{α}}.

To every limit α < λ we attach a club of α, e_{α}, satisfying e_{α} ∩ S = e_{α} ∩ S^{∗} = ∅, otp e_{α} = cfα and e_{α} contains only limit ordinals whenever α ∈ S^{∗}. Let \overline{C}_{0} = ⟨e_{δ} : δ ∈ S^{∗}⟩. Suppose that \overline{C}_{n} = ⟨c_{δ}^{n} : δ ∈ S⟩ is a bad candidate for the job, namely that there are a club E_{n} and a set X_{n} such that for every δ ∈ E_{n}∩S^{∗} the set {α ∈ nacc c_{δ}^{n} : α ∈ S(X_{n})∩E_{n}} is bounded below δ. (Surely, we may assume that E_{n} is as thin as we like — in particular that all its members are limits). Define \overline{C}_{n+1} by induction on δ: For every γ ∈ c_{δ}^{n}, we define c_{δ}^{n+1}∩(γ, min c_{δ}^{n}\(γ+1)) (where (γ, β) denotes, as usual, an open interval of ordinals), and we let c_{δ}^{n+1} = c_{δ}^{n}∪⋃{c_{δ}^{n+1}∩(γ, min c_{δ}^{n}\(γ+1)) : γ ∈ c_{δ}^{n}}. This is well defined, as every γ ∈ c_{δ}^{n} has a successor in c_{δ}^{n}. So denote by β the ordinal min c_{δ}^{n}\(γ+1), and let

(∗) c_{δ}^{n+1} ∩ (γ, β) = \{ \begin{matrix}c_{β}^{n+1} ∩ (γ, β) & if β ∈ S^{∗} \\ ∅ & if β ∈ S(X_{0}, ⋯, X_{n}) ∩ E_{n} \\ \{α : γ < α < β ∧ (∃ζ ∈ e_{β})(α = sup(ζ ∩ E_{n}))\} & otherwise \end{matrix}

Note that for the definition to be consistent, β ∈ c_{δ}^{n} must always be limit (and this is indeed the case).

4 Souslin 27.9.1998 3.1 Lemma: Suppose that \overline{C}_{n} is defined for n ≤ m. Then for For every n < m and δ ∈ S^{∗}:

(0) If α ∈ c_{δ}^{n} then β is a limit ordinal;

(1) c_{δ}^{n} is closed.
(2) c_{δ}^{n} ⊆ c_{δ}^{n+1}.
(3) If α ∈ S^{∗} ∩ acc c_{δ}^{n}, then c_{α}^{n} and c_{δ}^{n+1} ∩ α have a common end segment.
(4) If α ∈ c_{δ}^{n+1} ∩ S(X_{0}, ⋯, X_{n}) ∩ E_{n}, then α ∈ nacc c_{δ}^{n+1}.

Proof: : (2) is true by the definition of c_{δ}^{n+1} for every n and δ ∈ S^{∗}. (0), (1), (3) and (4) are proved by induction on n and δ.

For n = 0 we know that e_{δ} = c_{δ}^{0} is all limits and is closed, so (0) and (1) hold. (3) is vacuously true, because e_{δ} ∩ S^{∗} = ∅, and (4) is vacuously true because e_{δ} ∩ S = ∅.

For n + 1:

(0): Suppose α ∈ c_{δ}^{n+1}. If α ∈ c_{δ}^{n} then it is a limit ordinal by (0) and the induction hypothesis on n. If α ∉ c_{δ}^{n}, let γ = sup c_{δ}^{n} ∩ α. Because of (1) and the induction hypothesis γ < α. Let β = min c_{δ}^{n} \ (α + 1). If β ∈ S^{∗} then c_{δ}^{n+1} ∩ (γ, β) = c_{β}^{n+1} ∩ (γ, β). So α ∈ c_{β}^{n+1}, and by the induction hypotheses on β, α is a limit ordinal. If β ∉ S^{∗}, the c_{δ}^{n+1} ∩ (γ, β) = {α : γ < α < β, (∃ζ ∈ e_{β})(α = sup ζ ∩ E_{n})}. Therefore for some ζ ∈ e_{β} our α is sup(ζ ∩ E_{n}). Since E_{n} is a club, α ∈ E_{n}. But E_{n} is a club of limits, so α is limit.

(1) Suppose that α ∈ acc c_{δ}^{n+1}, and we wish to show α ∈ c_{δ}^{n+1}. If α ∈ acc c_{δ}^{n}, then because of (1) and the induction hypothesis on n α ∈ c_{δ}^{n} and (by (2)) α ∈ c_{δ}^{n+1}. Else, γ = sup α ∩ c_{δ}^{n} and β = min c_{δ}^{n} \ (a + 1), γ < α < β. If β ∈ S^{∗} then α ∈ acc c_{β}^{n+1}. By the induction hypothesis on β and (1), α ∈ c_{δ}^{n+1}. Otherwise, α is a limit of ⟨a_{i} : i < i^{∗}⟩ such that α_{i} = sup ζ_{i} ∩ E_{n} ∈ c_{δ}^{n+1}. So clearly α ∈ E_{n}. Let ζ^{∗} be the minimal in e_{β} above α. So α = sup ζ^{∗} ∩ E_{n}. Therefore α ∈ c_{δ}^{n+1}.

Before proving (3) we note: 3.2 Fact: Suppose γ ∈ c_{δ}^{n} and β = min c_{δ}^{n} \ (γ + 1). If β ∉ S^{∗} and α ∈ c_{δ}^{n+1} ∩ (γ, β) is a limit of c_{δ}^{n+1}, then α ∈ e_{β}.

Indeed, if α = sup{α(i) : i < i^{∗}}, where α(i) = sup ζ(i) ∩ E_{n} are elements in c_{δ}^{n+1}, α ∈ E_{n}. Therefore every ζ(i) < α (or else sup ζ(i) ∩ E_{n} ≥ α > α(i)). But ζ(i) ≥ α(i), so α is a limit of e_{β}. As α < β and e_{β} is closed, α ∈ e_{β}.⊣3.2

(3): Let α ∈ acc c_{δ}^{n+1} ∩ S^{∗}, and we wish to show that c_{δ}^{n+1} and c_{α}^{n+1} have a common end segment. If α ∈ acc c_{δ}^{n}, then by the induction hypothesis on n and (3), we know that c_{δ}^{n} and c_{α}^{n} have a common end segment; say they agree on the interval (α(0), α). This means in particular that for every γ ∈ c_{δ}^{n} ∩ (α(0), α), α ∈ c_{α}^{n} and min c_{δ}^{n} \ (γ + 1) = min c_{α}^{n} \ (γ + 1) =: β. Therefore also c_{δ}^{n+1} ∩ (γ, β) = c_{α}^{n+1} ∩ (γ, β), and consequently c_{δ}^{n+1} ∩ (α(0), α) = c_{α}^{n+1} ∩ (α(0), α). So assume that α ∉ acc c_{δ}^{n}. The first possibility is that α ∉ c_{δ}^{n} altogether. In this case let γ < α < β assume their traditional roles as the last ordinal of c_{δ}^{n} below α and the first above. If β ∈ S^{∗}, then by

5 Souslin 27.9.1998 the induction hypothesis on β we know that c_{β}^{n+1} and c_{α}^{n+1} have a common end segment; but c_{δ}^{n+1} ∩ (γ, β) = c_{β}^{n+1} ∩ (γ, β), so it follows that c_{δ}^{n+1} and c_{α}^{n+1} have a common end segment.

If β ∉ S^{∗}, then by the Fact above, α ∈ e_{β} — contradiction to e_{β} ∩S^{∗} is empty. So this subcase is non existent.

The last case is: α ∉ acc c_{δ}^{n} but α ∈ c_{δ}^{n}, or in short α ∈ nacc c_{δ}^{n}. Let γ be the last element of c_{δ}^{n} ∩ α. Then by (∗), c_{δ}^{n+1} ∩ (γ, α) = c_{α}^{n+1} ∩ (γ, α).

(4): Suppose that α ∈ S(X_{0}, ·, X_{n}) ∩ c_{δ}^{n+1}. We should see that α ∈ nacc c_{δ}^{n+1}. Let m ≤ n + 1 be the minimal such that α ∈ c_{δ}^{m}. It is enough to prove that α ∈ nacc c_{δ}^{m}, because by (∗) it is clear that if α ∈ S(X_{0}, ⋯, X_{m}) ∩ nacc c_{δ}^{m} then α will remain a non-accumulation point in c_{δ}^{m+1} (because nothing will be added in the interval below it). So without loss of generality we may assume that α ∈ c_{δ}^{n+1} \ c_{δ}^{n}. So denote by (γ, β), as usual, the unique minimal interval with end points in c_{δ}^{n} to which α belongs. First case: β ∈ S^{∗}. So α ∈ c_{β}^{n+1}; and by the induction hypothesis on β, α ∈ nacc c_{β}^{n+1}. So this case is done. Otherwise, β ∉ S^{∗}. So by the Fact above, if α were a limit of c_{δ}^{n+1}, it would be in e_{β}. But α ∈ S, and therefore cannot be in e_{β} by the very choice of e_{β}. Therefore α ∈ nacc c_{δ}^{n+1}. (This is where the non reflection of S is used in an essential way).⊣3.1 3.3 Claim: : There is some n < ω for which \overline{C}_{n} and ⟨A_{n}^{α} : α ∈ S⟩ are as required. Proof: : Suppose not. Let \overline{X}_{ω} = ⟨X_{n} : n < ω⟩. Let E = ∩_{n}E_{n} and E′ = acc (S(\overline{X}_{ω}) ∩ E). So E′ is a club. Pick some δ ∈ S^{∗} ∩ E′. For every n there is a bound below δ of the set {α ∈ nacc c_{δ}^{n} : α ∈ S(X_{n}) ∩ E_{n}}. As cfδ > ℵ_{0}, let α^{∗} < δ bound α(n) for all n. Let δ > β > α^{∗} be in S(\overline{X}_{ω}) ∩ E. So for every n, X_{n} ∩ β = A_{n}^{β} and β ∈ E_{n}. If β ∈ c_{δ}^{n} for some n, then by (4) β ∈ nacc c_{δ}^{n} — a contradiction to β > α(n). So β ∉ c_{δ}^{n} for all n. Therefore for every n we may define (γ(n), β(n)) as the minimal interval with ends in c_{δ}^{n} which contains β. 3.4 Claim: β(n + 1) < β(n). Proof: By its definition, β(n) ∈ nacc c_{δ}^{n}. In the case β(n) = δ^{∗} ∈ S^{∗}, there are clearly elements in c_{δ^{∗}}^{n+1} above β and below β(n), so the claim is obvious. The case β(n) ∈ S(X_{0}, ⋯, X_{n}) is impossible because of (4). In the remaining case, c_{δ}^{n+1}∩(γ(n), β(n)) = {α : γ(n) < α < β(n), (∃ζ ∈ e_{β(n)})(α = sup ζ ∩ E_{n})}. Let ζ^{∗} > β be in e_{β(n)}. As β ∈ E ⊆ E_{n}, sup ζ^{∗} ∩ E_{n} ≥ β. But the right hand side of this inequality belongs to c_{δ}^{n+1}, while β does not; therefore sup ζ^{∗} ∩ E_{n} > β. So we see that there are elements of c_{δ}^{n+1} in (β, β(n)), therefore the least of them, namely β(n+1) is smaller than β(n).⊣3.4

This is clearly a contradiction. We conclude that after finitely many steps, \overline{C}_{n+1} cannot be defined due to the lack of a counterexample. This means that \overline{C}_{n} and ⟨B_{α} : α ∈ S⟩ where B_{α} = A_{α}^{n} satisfy (i), (ii) and (iv). By (3) above, they satisfy (iii) as well. ⊣3.3

This shows that after finitely many corrections all the requirements are satisfied, and our theorem is proved.⊣3

4. Theorem: If the e_{δ} we pick in the proof of Theorem 3 satisfy the additional condition that

6 Souslin 27.9.1998 for every γ < λ the set {e_{δ} ∩ γ : α ∈ λ is limit} has cardinality smaller then λ, then the resulting good \overline{C} = ⟨c_{δ} : δ ∈ S^{∗}⟩ satisfies that for every γ < λ, |\{c_{δ} ∩ γ : δ ∈ S^{∗}\}| < λ Proof: Let γ < λ be given. We must show that |{c_{δ}^{n}∩γ : δ ∈ S^{∗}}| ≤ μ. Let N ≺ H(χ, ∈) for some large enough χ, |N| < λ, γ ⊆ N, γ ∈ N, {e_{α} ∩ γ : α < λ is limit} ⊆ N, ⟨e_{α} : ν < λ is limit⟩ ∈ N, and E_{n}, X_{n} ∈ N for every n.

We shall see that for every n and δ, c_{δ}^{n} ∩ γ ∈ N. Since |N| < λ, this is enough.

First we notice that if δ < γ then c_{δ} ∈ N and by elementarity also c_{δ}^{n} ∈ N for every n. Now we use induction on n to show that for every δ > γ, c_{δ}^{n} ∩ γ ∈ N. For n = 0: if δ > γ then c_{δ}^{0} ∩ γ = e_{δ} ∩ γ ∈ N by the assumptions on N. For n + 1 we use induction on δ. Suppose, then, that for all δ′ < δ, c_{δ′}^{n+1} ∩ γ ∈ N.

We need the easy 4.1 Fact: If (α_{0}, α_{1}) is a minimal interval of c_{δ}^{n} ∩ (γ + 1) then c_{δ}^{n+1} ∩ (α_{0}, α_{1}) ∈ N. Proof: By (∗) above, the definition of c_{δ}^{n+1} ∩ (α_{0}, α_{1}) depends only on e_{α_{1}}, E_{n} and (if case there is such) c_{α_{1}}^{n+1}. All these objects are in N, so also c_{δ}^{n+1} ∩ (α_{0}, α_{1}) ∈ N.⊣4.1

Denote γ(δ) = sup c_{δ}^{n} ∩ γ. So γ(δ) ≤ γ. If γ(δ) = γ, then c_{δ}^{n+1} ∩ γ = c_{δ}^{n} ∩ γ ∪ ⋃_{I} c_{δ}^{n+1} ∩ I where I runs over all minimal intervals of c_{δ}^{n} ∩ (γ + 1). So by the Fact above we are done. Else, γ(δ) < γ. In this case define β(δ) = min c_{δ}^{n} \ γ. If β(δ) = γ then again we are done by the Fact. The remaining case is γ(δ) < γ < β(δ). By the same Fact, c_{δ}^{n+1} ∩ γ(δ) ∈ N. If β(δ) ∈ S^{∗}, then c_{δ}^{n+1} ∩ (γ(δ), γ) = c_{β(γ)}^{n+1} ∩ (γ(δ), γ). By the induction hypothesis, and since β(δ) < δ, the latter set is in N, and we are done. If β(δ) ∉ S^{∗}, then either nothing is added into (γ(δ), β(δ)) (when β(δ) ∈ S(X_{0}, ⋯, X_{n})), or c_{δ}^{n+1}∩(γ(δ), β(δ)) = {α : γ(δ) < α < β(d) (∃ζ ∈ e_{β(δ)})(α = sup E_{n}∩ζ)}. So in this definition N might not know who β(δ) is, but e_{β(δ)} ∩ γ ∈ N. Therefore, denoting by α^{∗} the last member in E_{n} ∩ γ, we can determine in N the set c_{δ}^{n+1} ∩ α^{∗}. As to whether α^{∗} itself is in this set or not, we need knowledge which is not available in N, but who cares, as long as both possibilities are in N. ⊣4

5. Corollary: If there is a non-reflecting stationary set S ⊆ {α < μ^{+} : cfα < μ}, and 2^{μ} = μ^{+}, μ^{<μ} = μ, then there is a μ-complete Souslin tree on μ^{+}.

6. Remark: This improves the result by Gregory in [G]. Proof: It is known (see [G] 2.1) that if S ⊆ {δ ∈ μ^{+} : cfδ < μ} is stationary, then μ = μ^{<μ} implies ♢(S). As S is non reflecting, we can, for every limit α < μ^{+}, choose a closed set e_{α}, α = sup e_{α} and otp e_{α} = cfα such that e_{α} ∩ S = ∅. μ = μ^{<μ} implies that for every γ < μ^{+} the set {e_{α} ∩ γ : α < λ, α is limit} is of cardinality at most μ. Use Theorem 3 and Theorem 4 to obtain the assumptions of Theorem 2, S being the given non reflecting stationary set and S^{∗} being {δ < λ : cfδ = μ}. By Theorem 2 there is an μ-complete Souslin tree on μ^{+}.⊣5

7. Problem: (1) Can the existence of such a tree be proved in ZF C + GCH?

(2) Can a Souslin tree on ℵ_{2} be constructed from GCH and two stationary sets, each composed of ordinals of countable cofinality, which do not reflect simultaneously? By [Mg] this would raise the consistency strength of GCH +SH(ℵ_{2}) to the consistency of the existence of a weakly compact cardinal.

References

[AbShSo] U. Abraham, S. Shelah and R. M. Solovay Squares with diamonds and Souslin trees with special squares, Fundamenta Mathematicae vol. 127 (1986) pp. 133–162

[De] K. J. Devlin, Aspects of Constructibility, Lecture Notes Math. 354, 240 pp., 1973.

[DeJo] K. J. Devlin and H. Johnsbråten The Souslin Problem, Lecture Notes Math. 405, 132 pp., 1974.

[J] R. Björg Jensen, The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229–308.

[Je] T. Jech, Set Theory, Academic press, 1978.

[G] J. Gregory, Higher Souslin Trees and the Generalized Continuum Hypotheses, The Journal of Symbolic Logic, Volume 41, Number 3, 1976, pp. 663–671.

[LvSh 104] R. Laver and S. Shelah, The ℵ_{2}-Souslin hypothesis, Trans. of A.M.S., vol. 264 (1981), pp. 411–417.

[Mg] M. Magidor, Reflecting stationary sets, The Journal of Symbolic Logic vol 47, number 4, 1982 pp.755–771.

[Sh e] S. Shelah, Non Structure Theory, Oxford University Press, accepted. [Sh-g] S. Shelah, Cardinal Arithmetic, submitted to Oxford University Press. [Sh 365] S. Shelah, There are Jonsson algebras in many inaccessible cardinals, a chapter in

Cardinal Arithmetic.

[ShSt 154] S. Shelah and L. Stanley, Generalized Martin Axiom and the Souslin Hypothesis for higher cardinality, Israel J. of Math, vol. 43 (1982) 225–236.

[ShSt 154a] S. Shelah and L. Stanley, Corrigendum to ”Generalized Martin’s Axiom and Souslin Hypothesis for Higher cardinality, Israel J. of Math vol. 53 (1986) 309–314.

[ShSt 279] S. Shelah and L. Stanley, Weakly compact cardinals and non special Aronszajn trees , Proc. A.M.S., vol. 104 no. 3 (1988) 887-897.